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###### 日期：2020-10-24 10:40

Computer Science 272: Data Structures and Algorithms Page 1 of 2

Assignment 3

Answer all questions – maximum 100 marks. You must score at least 50 to pass the assignment.

1. (5 marks) Illustrate that the nodes of any AVL tree T can be colored “red” and “black” so that T

becomes a red-black tree.

2. (5 + 10 = 15 marks) Illustrate that via AVL single rotation, any binary search tree T1 can be

transformed into another search tree T2 (with the same items) (5 marks).

Give an algorithm to perform this transformation using O(N log N) rotation on average (10 marks).

3. (10 + 2 = 12 marks) Suppose you are given two sequences S1 and S2 of n elements, possibly

containing duplicates, on which a total order relation is defined. Describe an efficient algorithm for

determining if S1 and S2 contain the same set of elements (10 marks).

Analyze the running time of this method (2 marks).

4. (5 + 8 = 13 marks) Given sequence 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, sort the sequence using the following

algorithms, and illustrate the details of the execution of the algorithms:

a. (5 marks) merge-sort algorithm.

b. (8 marks) quick-sort algorithm. Choose a partitioning strategy you like to pick a pivot element

from the sequence. Analyze how different portioning strategies may impact on the performance

of the sorting algorithm.

5. (4 + 4 + 7 + 10 = 25 marks) Given the graph shown below, answer the following questions:

a. (4 marks) Illustrate the sequence of vertices of this graph visited using depth-first search traversal

starting at vertex g.

b. (4 marks) Illustrate the sequence of vertices of this graph visited using breadth-first search

traversal starting at vertex b.

respectively, for this graph. What are the advantages and disadvantages of those two

representations?

d. (10 marks) Describe an algorithm to find in the graph a path illustrated below that goes through

every edge exactly once in each direction.

Computer Science 272: Data Structures and Algorithms Page 2 of 2

6. (10 marks) Exercise 9.7. Why does the method remove(x) in the RedBlackTree implementation

perform the assignment u:parent = w:parent? Shouldn’t this already be done by the call to

splice(w)?

7. (10 marks) Exercise 10.8. Implement the remove(u) method, that removes the node u from a

MeldableHeap. This method should run in O(log n) expected time.

8. (10 marks) Exercise 11.12. Prove that a binary tree with k leaves has height at least log k.